Now it is straightforward to obtain the result that for any two rationals $a,b$, there is (at least) countably infinitely many rational number between; it can formally be obtained by induction, but one see that one can just repeat the process and succesively obtain rationals between given ones.

There is also a nice geometric interpretation of this method. If we imagine rationals on a number line, then arithmetic mean $\frac{a+b}{2}$ lands exactly in the center of the interval $[a,b]$ (that is, it halves it).